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A white hole is the theorized time reversal of a black hole. The event horizon of a black hole attracts matter, so event horizon of a white hole ejects matter even though the white hole itself still attract matter. The main difference between the two is the action of the event horizon.
The event horizon of a black hole will engulf every particle of matter that it encounters; however, a white hole shrinks away from any and all matter so that nothing ever crosses the event horizon. The matter ends up scattered when the hole collapses from its constant recession. Using quantum mechanics, Steven Hawking demonstrated that a black hole emits Hawking radiation and can come to thermal equilibrium. That same thermal equilibrium stays unchanged in time reversal. So, the reverse of a black hole in thermal equilibrium is a black hole in thermal equilibrium; meaning that a hole, black or white, is the same thing.
The concept of a white hole only appears as part of the vacuum solution to Einstein’s field equations that are used to describe a Schwarzschild wormhole. A wormhole is a black hole on one end, drawing in matter, and a white hole on the other to emit matter. Schwarzschild wormholes are unstable. They collapse as soon as they form. Also, wormholes are only a solution to the Einstein field equations in a vacuum where no matter interacts with the hole. Real black holes are formed by the collapse of stars, but white holes shrink from matter so they could not exist in connection with true black holes because the presence of matter would cause them to collapse.
A white hole is only a concept for higher levels of thinking. No one has every observed one and no one probably ever will. A few scientist think that a white hole could be part and parcel of a concept called a Fecund universe.
We’ve also recorded an episode of Astronomy Cast all about White Holes. Listen here, Episode 31: String Theory, Time Travel, White Holes, Warp Speed, Multiple Dimensions and Before the Big Bang.